The mathematical expression in the image is:

$(2^3)^4 \cdot (2^2)^{-5}$

To simplify this, we will apply the laws of exponents.

Step-by-Step Solution:

1. Simplify each term: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$ $(2^2)^{-5} = 2^{2 \cdot (-5)} = 2^{-10}$

2. Multiply the powers: When multiplying terms with the same base, we add the exponents: $2^{12} \cdot 2^{-10} = 2^{12 + (-10)} = 2^2$

3. Final result: $2^2 = 4$

Thus, the result of the expression $(2^3)^4 \cdot (2^2)^{-5}$ is $4$.

Would you like further clarification or details on any of the steps?

Here are 5 related questions to expand your understanding:

1. What are the rules for multiplying powers with the same base?
2. How do you simplify expressions with negative exponents?
3. Can you apply the same exponent rules to expressions with different bases?
4. What happens when exponents are zero?
5. How do you divide powers with the same base?

Tip: Always remember to apply the exponent rules carefully, especially when dealing with negative exponents!